Dehn filling of a Hyperbolic 3-manifold Maria Trnkov Department of - - PowerPoint PPT Presentation

▶

Jun 20, 2023 375 likes •543 views

Dehn filling of a Hyperbolic 3-manifold Maria Trnkov Department of Mathematics University of California, Davis In collaboration with Matthias Goerner, Neil Hoffman, Robert Haraway TOPOSYM, July 29, 2016 1 /15 Maria Trnkov Dehn filling of

SLIDE 1

Dehn filling of a Hyperbolic 3-manifold

Maria Trnková

Department of Mathematics University of California, Davis

In collaboration with Matthias Goerner, Neil Hoffman, Robert Haraway

TOPOSYM, July 29, 2016

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

1/15

SLIDE 2

Plan of the talk

1 Background of hyperbolic 3-manifolds. 2 Dehn filling. 3 Dehn parental test. Maria Trnková Dehn filling of a Hyperbolic 3-manifold

2/15

SLIDE 3

Background

Definition: A hyperbolic 3-manifold is a quotient H3/Γ of three-dimensional hyperbolic space H3 by a subgroup Γ of hyperbolic isometries PSL(2, C) acting freely and properly discontinuously. The subgroup Γ is isomorphic to the fundamental group π1(M).

Theorem (Mostow-Prasad Rigidity, ’74)

If M1 and M2 are complete finite volume hyperbolic n-manifolds, n > 2, any isomorphism of fundamental groups ϕ : π1(M1) → π1 (M2) is realized by a unique isometry. Geometric invariants (volume, geodesic length) are topological invariants. Thurston, Jorgensen (1977) gave classification of finite volume hyperbolic 3-manifolds by their volume.

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

3/15

SLIDE 4

Background

M is a complete finite volume hyperbolic 3-manifold:

closed
cusped

Dirichlet domains of closed and cusped hyperbolic 3-manifolds from SnapPy Maria Trnková Dehn filling of a Hyperbolic 3-manifold

4/15

SLIDE 5

Background

Every element γ ∈ Γ corresponds to a closed geodesic g ⊂ M. Every preimage of g in H3 is preserved by γ or its conjugates. Definition: Complex length l(γ) of a closed geodesic g in a hyperbolic 3-manifold is a number λ + iθ, λ is a geodesic’s length and a minimal distance of transformation γ, θ is the angle of rotation incurred by traveling once around γ, defined modulo 2π. Definition: Length spectrum L(M)

f a hyperbolic 3-manifold is the set
f complex length of all closed geo-

desics in M taken with multiplicities: L(M) = {l(γ)|∀γ ∈ Γ} ⊂ C. It is a discrete ordered set.

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

5/15

SLIDE 6

Dehn Filling

M is a complete finite volume hyperbolic 3-manifold:

closed
cusped

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

6/15

SLIDE 7

Dehn Filling

Drilling in dimension 2:

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

7/15

SLIDE 8

Dehn Filling

M - complete finite volume hyperbolic 3-manifold, ∂M = ⊔ Ti Dehn filling of M - “compactification”. Glue back solid torus with a Dehn twist. Result not always a manifold.

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

8/15

SLIDE 9

Dehn Filling

Framing of each Ti: set of meridians and longitudes (µ, λ). Definition: Slope is an isotopy class of unoriented essential simple closed curves in the boundary of M. Slope is identified with element of Q ∪ ∞ via p/q ↔ ±(pµ + qλ).

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

9/15

SLIDE 10

Dehn Filling

Theorem (Thurston’s Dehn Surgery Theorem, 1970’s)

Let M - compact, orientable 3-manifold, ∂M = ⊔ Ti - finite number of tori components, interior of M - admits complete, finite volume hyperbolic metrics. Then ALL BUT A FINITE number of filling curves on each Ti give a closed 3-manifold with hyperbolic structure (otherwise we have “exceptional curves”). Question: How many exceptional fillings a manifold M has? Answer: At most 10 for 1-cusped manifolds (M.Lackenby - R.Meyerhoff, 2008).

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

10/15

SLIDE 11

Dehn Parental Test

M, N - orientable 3-manifolds, admit complete hyperbolic metrics of finite volume on their interiors. Question: Is N a Dehn filling of M?

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

11/15

SLIDE 12

Dehn Parental Test

C.Hodgson - S.Kerckhoff (2008) described the first practical method for determining Dehn filling heritage.

Theorem (R.Haraway, 2015)

Let M, N be orientable 3-manifolds admitting complete hyperbolic metrics of finite volume on their interiors. Let ∆V = V ol(M) − V ol(N). N is a Dehn filling of M if and only if either: N is a Dehn filling of M along a slope c of normalized length L(c) ≤ 7.5832, or N has a closed simple geodesic γ of length l(γ) < 2.879∆V and N is a Dehn filling of M along a slope c such that 4.563/∆V ≤ L2(c) ≤ 20.633/∆V .

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

12/15

SLIDE 13

Dehn Parental Test

Dehn parental test for hyperbolic 3-manifolds reduces to rigorous calculations of volume (HIKMOT in Python, 2013), length spectra (Ortholength.nb, D.Gabai-M.T., 2012), cusp area, slope length (fef.py by B.Martelli-C.Petronio-F.Roukema, 2011, K.Ichihara-H.Masai, 2013), isometry test (SnapPea by J.Weeks). Work in progress: write a rigorous algorithm for length spectra in Python using interval arithmetic. combine all existing programs to perform Dehn parental test as one command in SnapPy.

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

13/15

SLIDE 14

Conclusion

Dehn parental test: allows to determine Dehn filling heritage between two hyperbolic 3-manifolds can be verified rigorously with computer programs.

Maria Trnková Dehn filling of a Hyperbolic 3-manifold

14/15

SLIDE 15

Conclusion

Dehn parental test: allows to determine Dehn filling heritage between two hyperbolic 3-manifolds can be verified rigorously with computer programs.

THANK YOU!

Maria Trnková Dehn filling of a Hyperbolic 3-manifold